BOP tuning: Difference between revisions

Line 81:

'''Theorem:''' to minimize the <math>1/(nd)^s</math> weighted error on all rationals, one needs only minimize the <math>1/p^s</math> weighted error on the primes.

== Extension to ~~"Good"~~ Subgroups ==

== Extension to Arbitrary Subgroups ==

The above proof was only for prime limits. However, the above result can be extended easily to certain "nice" subgroups with a basis of prime powers, such as 2.9.7.11, with the only caveat being that we want to make sure we directly weight each prime power <math>p^n</math> as <math>p^{ns}</math>, rather than giving it the naive weighting of <math>\text{sopfr}^s(p^n)</math>. ~~More work is needed to derive~~ a corresponding result for ~~non-nice subgroups~~.

The above proof was only for prime limits. However, the above result can be extended easily to certain "nice" subgroups with a basis of prime powers, such as 2.9.7.11, with the only caveat being that we want to make sure we directly weight each prime power <math>p^n</math> as <math>p^{ns}</math>, rather than giving it the naive weighting of <math>\text{sopfr}^s(p^n)</math>.

In general, a corresponding result can be derived for any arbitrary subgroup; the max 1/(n/d)^s weighted error will be on some relatively simple interval, so that one only needs to check a sufficiently small set of intervals (though not necessarily the primes).

= Proof of Optimality Even With Extra "Inconsistent" Mappings =

@@ Line 81: / Line 81: @@
 '''Theorem:''' to minimize the <math>1/(nd)^s</math> weighted error on all rationals, one needs only minimize the <math>1/p^s</math> weighted error on the primes.
-== Extension to "Good" Subgroups ==
+== Extension to Arbitrary Subgroups ==
-The above proof was only for prime limits. However, the above result can be extended easily to certain "nice" subgroups with a basis of prime powers, such as 2.9.7.11, with the only caveat being that we want to make sure we directly weight each prime power <math>p^n</math> as <math>p^{ns}</math>, rather than giving it the naive weighting of <math>\text{sopfr}^s(p^n)</math>. More work is needed to derive a corresponding result for non-nice subgroups.
+The above proof was only for prime limits. However, the above result can be extended easily to certain "nice" subgroups with a basis of prime powers, such as 2.9.7.11, with the only caveat being that we want to make sure we directly weight each prime power <math>p^n</math> as <math>p^{ns}</math>, rather than giving it the naive weighting of <math>\text{sopfr}^s(p^n)</math>.
+In general, a corresponding result can be derived for any arbitrary subgroup; the max 1/(n/d)^s weighted error will be on some relatively simple interval, so that one only needs to check a sufficiently small set of intervals (though not necessarily the primes).
 = Proof of Optimality Even With Extra "Inconsistent" Mappings =